Tests of Equal Forecast Accuracy and Encompassing for Nested Models

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Abstract
Ashley, Granger, and Schmalensee (1980) and Diebold and Mariano (1995) suggest that forecast comparisons may be used to examine Granger causality. According to Ashley et al., if forecasts of y based on a VAR model in x and y are superior to those based on an AR model for y , then x carries information about y and hence causes y . Thus, this combination of tests can be viewed as a test for Granger causality. This paper examines both asymptotic and finite-sample properties of tests for equal accuracy and encompassing in the VAR vs. AR settings. The statistics considered include the Diebold-Mariano test for equal accuracy and the Chong and Hendry (1986) and Harvey et al. (1998) tests for encompassing. Because the Diebold-Mariano and Harvey et al. tests are designed for non-nested forecasts, they are asymptotically invalid when applied to forecasts from nested models. Asymptotically valid versions of the tests are also considered. Valid versions include an out-of-sample F -type statistic and a version of the Diebold-Mariano t -statistic developed in McCracken (1999), as well as a variant of the Harvey et al. encompassing statistic developed in this paper. The asymptotic tests developed by McCracken are extended to multi-step forecast horizons. Monte Carlo simulations show that using asymptotically invalid tests can produce misleading inferences in small samples. The simulations also indicate that out-of-sample F -type and encompassing tests can be more powerful than standard F -tests of causality.
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